Mathematical formulas

Algebra

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Algebra
Complex numbers

Section 13

Complex numbers

Rectangular form

fundamentalviz

i^2=-1;\quad z=a+bi

Cartesian representation of a complex number.

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fundamentalviz

\bar z=a-bi;\quad z\bar z=|z|^2

Reflects a complex number across the real axis.

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fundamentalviz

|z|=\sqrt{a^2+b^2}

Distance from the complex number to the origin.

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Polar and exponential form

intermedioviz

z=r(\cos\theta+i\sin\theta)=re^{i\theta}

Represents a complex number by magnitude and angle.

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Euler's formula

intermedioviz

e^{i\theta}=\cos\theta+i\sin\theta

Connects complex exponentials with trigonometry.

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intermedioviz

(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)

Raises complex numbers in polar form to a power.

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intermedioviz

w_k=r^{1/n}e^{i(\theta+2\pi k)/n}

The roots are distributed uniformly on a circle.

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